592 



or, using tlie substitution 



X, = Vb^ 



M, 



1/2 f (1 





'^J^c'(b'^y,rJd-/i.nn'Xcos''A= I 



1/1 + 



(26) 



5jr 1/2 f (1—6') 



c' (b'^y.r 



250 l/l+t' 



Formula (25) was chosen with a view of obtaining this latter 

 result for il/,, which facilitates the further calculations. If a new 

 variable i] is introduced, determined bj- the formula: 



V = 



•lb 



(it appears from this formula (hat i/ has the value on the tangent 

 at the ellipse at the point D, and takes the value 1 on the tangent 

 at C), then equation (26) can be written: 



M,=:An^{\-ny^A.,{n). (27) 



Here A is a factor independent of the variable ?;. 



If we imagine a great number of these vortices to be present, 

 all of them having the same dimensions and lying between the 

 same tangents parallel to the axis of x, (comp. fig. 3), the amount 

 contributed bj them to the value of uv will be proportional to the 

 function represented by (27) '). 



Fig. 3. 



Tlie integral of the quantity M, taken over the entire area of 

 the vortex amounts to: 



' 63 1 + f' 



(28) 



') Other types of motion may lead to the same form of the function determin- 

 ing Jfi ; for instance we may lake tlie motion defined by the current function 



W r= rf (1 — if) (el— 'i cos ux — €''• sin ax) 

 for values of /; between and I, so that the components of the velocity have 

 tlie values : 



